A Jordan algebra for hydrogen atom and space-time symmetries

It has been realized long ago that the 15 dimensional conformal group, extending the Poincaré group with dilations and conformal inversions is a symmetry of the Maxwell equations. Conformal action yields a special mass-zero representation preserving the causal structure of Minkowski spacetime. On the other hand in the seventies in the works of Barut and others the conformal group emerged as a dynamical symmetry of the hydrogen atom. In this note we show that the conformal symmetry of Minkowski spacetime and the hydrogen atom dynamical symmetry are actually the coordinate and momentum space representation of the conformal group of the Euclidean Jordan algebra of Pauli matrices.